PORISM

PORISM, in Geometry, is a name given by the ancient geometers to two classes of mathematical propositions. Euclid applies this name to propositions, which are involved in others which he is professedly investigating, and which, although they do not form his principal object, are obtained along with it, as is expressed by their name porismata, or acquisitions. Such propositions are now called corollaries. But he gives the same name, by way of eminence, to a particular class of propositions which he collected in the course of his researches, and selected from amongst many others, on account of their great subservience to the business of geometrical investigation in general. These propositions were so named by him, either from the way in which he discovered them, that is to say, whilst he was investigating something else, so that they might be considered as gains or acquisitions, or from their utility as steps in the investigation. In this sense they are porismata; for porisma (from which, according to Proclus, the term is derived) signifies both to investigate and to acquire by investigation. These propositions formed a collection in three books, which was familiarly known to the ancient geometers by the name of Euclid's porisms; and Pappus of Alexandria says, that it was a most ingenious collection of many things conducive to the analysis or solution of the most difficult problems, and which afforded great delight to those who were able to understand and investigate them.

Unfortunately for mathematical science, this valuable collection is now lost, and it still remains a doubtful question in what manner the ancients conducted their researches upon this curious subject. We have, however, reason to believe that their method was both comprehensive and excellent; for their analysis led them to many profound discoveries, and was restricted by the severest logic. The only account we have of this class of geometrical propositions, is in a fragment of Pappus, in which he attempts a general description of them, as a set of mathematical propositions distinguishable in kind from all others; but of this description nothing remains, except a criticism on a definition of the term given by some geometers, namely, "A Porism is that which is deficient in hypothesis from a local theorem," and which he finds fault with, as defining the porisms only by an accidental circumstance. Pappus also gives an account of Euclid's porisms; but the enunciations are so extremely defective, at the same time that they refer to a figure now lost, that Dr. Halley confesses the fragment in question to be beyond his comprehension.

The high encomiums pronounced by Pappus on these propositions have excited the curiosity of the greatest geometers of modern times, who have attempted to discover their nature and the manner of investigating them. Fermat gave a few propositions, which have been published in his Opera (Toulouse, 1679), and Bullialdus, in a tract entitled Exercitationes Geometricae Tres (Paris, 1657), attempted the same thing, but with less success. Albert Girard, at a still earlier period, announced that he had restored the whole of the three books of Euclid, but it does not appear that this part of his works was ever published.¹

¹ At length Dr. Simson, Professor of Mathematics in the University of Glasgow, was so fortunate as to succeed in

Porism. restoring the porisms of Euclid. In the preface to his treatise De Porismatibus, he gives the following account of his progress and of the obstacles he encountered: "Postquam vero apud Pappum legeram Porismata Euclidis Collectionem fuisse artificiosissimam multarum rerum, quæ spectant ad analysin difficiliorum et generalium problematum, magno desiderio tenebar, aliquid de iis cognoscendi; quare sæpius et multis variisque viis tum Pappi propositionem generalem, mancans et imperfectum, tum primum lib. 1. Porisma, quod, ut dictum fuit, solum ex omnibus in tribus libris integrum adhuc manet, intelligere et restituere conabar; frustra tamen, nihil enim proficiebam. Cumque cogitationes de hac re multum mihi temporis consumpsissent, atque tandem molestæ admodum evaserint, firmiter animum induxi hæc nunquam in posterum investigare; præsertim cum optimus Geometra Halleius spem omnem de iis intelligendis abjecisset. Unde quoties menti occurrebant, toties eas arcebam. Postea tamen accidit ut improvibus et propositi inmemorem invaserint, neque detinerent donec tandem lux quedam effulserit quæ spem mihi faciebat inveniendi saltem Pappi propositionem generalem, quam quidem multa investigatione tandem restitui."

Dr. Simson's Restoration has every appearance of being just. All the lemmas which Pappus has given for the better understanding of Euclid's propositions are equally applicable to those of Dr. Simson, which are found to differ from local theorems precisely as Pappus affirms those of Euclid to have done. They require a particular mode of analysis, and are of immense service in geometrical investigation.

Whilst Dr. Simson was employed in this inquiry, he carried on a correspondence upon the subject with the late Dr. Matthew Stewart, Professor of Mathematics in the University of Edinburgh; who, besides entering into Dr. Simson's views, and communicating to him many curious porisms, pursued the same subject in a new and very different direction. He published the result of his inquiries in 1746, under the title of General Theorems, not wishing to give them any other name, lest he might appear to anticipate the labours of his friend and former preceptor. The greater part of the propositions contained in that work are porisms, but without demonstration; and those who wish to investigate one of the most curious subjects in geometry, will there find abundance of materials, and an ample field for discussion.

Dr. Simson defines a porism to be "a proposition, in which it is proposed to demonstrate, that one or more things are given, between which, and every one of innumerable other things not given, but assumed according to a given law, a certain relation, described in the proposition, is to be shewn to take place."

This definition is somewhat obscure, but will be plainer if expressed thus: "A porism is a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions." This definition agrees with Pappus's idea of these propositions, as far at least as they can be understood from the fragment already mentioned; for the propositions here defined, like those which he describes, are, strictly speaking, neither theorems nor problems, but of an intermediate nature between both. They neither simply enunciate a truth to be demonstrated, nor propose a question to be resolved, but are affirmations of a truth in which the determination of an unknown quantity is involved. In as far, therefore, as they assert that a certain problem may become indeterminate, they are of the nature of theorems; and, in as far as they seek to discover the condition by which that is brought about, they are of the nature of problems.

In order to give our readers a clear idea of the subject of porisms, we shall consider them in the way in which it is probable they occurred to the ancient geometers in the

course of their researches. This will at the same time shew Porism. the nature of the analysis peculiar to them, and their great use in the solution of problems.

It appears to be certain, that it has been the solution of problems which, in all states of the mathematical sciences, has led to the discovery of geometrical truths. The first mathematical inquiries, in particular, must have occurred in the form of questions, where something was given, and something required to be done; and by the reasoning necessary to answer these questions, or to discover the relation between the things given and those to be found, many truths were suggested, which came afterwards to be the subject of separate demonstrations. The number of these was the greater, because the ancient geometers always undertook the solution of problems, with a scrupulous and minute attention, insomuch that they would scarcely suffer any of the collateral truths to escape their observation.

Now, as this cautious manner of proceeding was not better calculated to avoid error than to lay hold of every collateral truth connected with the main object of inquiry, these geometers soon perceived, that there were many problems which in certain cases would admit of no solution whatever, in consequence of a particular relation existing amongst the quantities which were given. Such problems were said to become impossible; and it was soon perceived, that this always happened when one of the conditions of the problem was inconsistent with the rest. Thus, when it was required to divide a line, so that the rectangle contained by its segments might be equal to a given space, it was found that this was possible only when the given space was less than the square of half the line; for when it was otherwise, the two conditions defining, the one the magnitude of the line, and the other the rectangle of its segments, were inconsistent with each other. Such cases would occur in the solution of the most simple problems; but if they were more complicated, it must have been remarked, that the constructions would sometimes fail, for a reason directly contrary to that just now assigned. Cases would occur, where the lines, which by their intersection were to determine the thing sought, instead of intersecting each other as they did commonly, or of not meeting at all, as in the above mentioned case of impossibility, would coincide with one another entirely, and of course leave the problem unresolved. It would appear to geometers upon a little reflection, that since, in the case of determinate problems, the thing required was determined by the intersection of the two lines already mentioned, that is, by the points common to both; so in the case of their coincidence, as all their parts were in common, every one of these points must give a solution, or, in other words, the solutions must be indefinite in number.

Upon inquiry, it would be found that this proceeded from some condition of the problem having been involved in another, so that, in fact, the two formed but one, and thus there was not a sufficient number of independent conditions to limit the problem to a single or to any determinate number of solutions. It would soon be perceived, that these cases formed very curious propositions of an intermediate nature between problems and theorems; and that they admitted of being enunciated in a manner peculiarly elegant and concise. It was to such propositions that the ancients gave the name of porisms.

This deduction requires to be illustrated by an example. Suppose, therefore, that it were required to resolve the following problem. A circle ABC, (fig. 1), a straight line DE, and a point F, being given in position, to find a point G in the straight line DE such, that GF, the line drawn from it to the given point, shall be equal to GB, the line drawn from it touching the given circle.

Suppose G to be found, and GB to be drawn touching the given circle ABC in B, let H be its centre, join HB, and let HD be perpendicular to DE. From D draw DL,

touching the circle ABC in L, and join HL; also from the centre G, with the distance GB or GF, describe the circle BKF, meeting HD in the points K and K'. It is evident that HD and DL are given in position and magnitude: also because GB touches the circle ABC, HBG is a right angle; and since G is the centre of the circle BKF, HB touches that circle, and consequently $HB^2$ or $HL^2\; =\; KH\; \backslash times\; HK\text{'}$; but because $KK\text{'}$ is bisected in D, $KH\; \backslash times\; HK\text{'}\; +\; DK^2\; =\; DH^2$, therefore $HL^2\; +\; DK^2\; =\; DH^2$. But $HL^2\; +\; LD^2\; =\; DH^2$, therefore $DK^2\; =\; DL^2$ and $DK\; =\; DL$. But DL is given in magnitude, and consequently K is a given point. For the same reason K' is a given point, therefore the point F being given in position, the circle KFK' is given in position. The point G, which is its centre, is therefore given in position, which was to be found. Hence this construction:

Having drawn HD perpendicular to DE, and DL touching the circle ABC, make DK and DK' each equal to DL, and find G the centre of the circle described through the points KFK'; that is, let FK' be joined and bisected at right angles by MN, which meets DE in G, G will be the point required; or it will be such a point, that if GB be drawn touching the circle ABC, and GF to the given point, GB is equal to GF.

The synthetical demonstration is easily derived from the preceding analysis; but it must be remarked, that in some cases this construction fails. For, first, if F fall anywhere in DH, as at F', the line MN becomes parallel to DE, and the point G is nowhere to be found; or, in other words, it is at an infinite distance from D. This is true in general; but if the given point F coincide with K, then MN evidently coincides with DE; so that, agreeable to a remark already made, every point of the line DE may be taken for G, and will satisfy the conditions of the problem; that is to say, GB will be equal to GK, wherever the point G is taken in the line DE: the same is true if F coincide with K. Thus we have an instance of a problem, and that too a very simple one, which, in general, admits but of one solution; but which, in one particular case, when a certain relation takes place among the things given, becomes indefinite, and admits of innumerable solutions. The proposition which results from this case of the problem is a porism, and may be thus enunciated:

"A circle ABC being given by position, and also a straight line DE, which does not cut the circle, a point K may be found, such, that if G be any point whatever in DE, the straight line drawn from G to the point K shall be equal to the straight line drawn from G touching the given circle ABC."

The problem which follows, appears to have led to the discovery of many porisms.

A circle ABC (fig. 2), and two points D, E, in a diameter

of it being given, to find a point F in the circumference of the given circle, from which, if straight lines be drawn to the given points E, D, these straight lines shall have to one another the given ratio of $\backslash alpha$ to $\backslash beta$, which is supposed to be that of a greater to a less. Suppose the problem resolved, and that F is found, so that FE has to FD the given ratio of $\backslash alpha$ to $\backslash beta$; produce EF towards G, bisect the angle EFD by FL, and DFG by FM: therefore $EL\; :\; LD\; ::\; EF\; :\; FD$, that is, in a given ratio, and since ED is given, each of the segments EL, LD, is given, and the point L is also given; again, because DFG is bisected by FM, $EM\; :\; MD\; ::\; EF\; :\; FD$, that is, in a given ratio, and therefore M is given. Since DFL is half of DFE, and DFM half of DFG, therefore LFM is half of (DFE + DFB), that is, the half of two right angles, therefore LFM is a right angle; and since the points L, M, are given, the point F is in the circumference of a circle described upon LM as a diameter, and therefore given in position. Now the point F is also in the circumference of the given circle ABC, therefore it is in the intersection of the two given circumferences, and therefore is found. Hence this construction: Divide ED in L, so that EL may be to LD in the given ratio of $\backslash alpha$ to $\backslash beta$, and produce ED also to M, so that EM may be to MD in the same given ratio of $\backslash alpha$ to $\backslash beta$; bisect LM in N, and from the centre N with the distance NL, describe the semicircle LFM; and the point F, in which it intersects the circle ABC, is the point required.

The synthetical demonstration is easily derived from the preceding analysis. It must, however, be remarked, that the construction fails when the circle LFM falls either wholly within or wholly without the circle ABC, so that the circumferences do not intersect; and in these cases the problem cannot be solved. It is also obvious that the construction will fail in another case, viz. when the two circumferences LFM, ABC, entirely coincide. In this case, it is farther evident, that every point in the circumference ABC will answer the conditions of the problem, which is therefore capable of numberless solutions, and may, as in the former instance, be converted into a porism. We are now to inquire, therefore, in what circumstances the point L will coincide with A, and also the point M with C, and of consequence the circumference LFM with ABC. If we suppose that they coincide, $EA\; :\; AD\; ::\; \backslash alpha\; :\; \backslash beta\; ::\; EC\; :\; CD$, and $EA\; :\; EC\; ::\; AD\; :\; CD$, or by conversion, $EA\; :\; AC\; ::\; AD\; :\; CD\; -\; AD\; ::\; AD\; :\; 2DO$, O being the centre of the circle ABC; therefore, also, $EA\; :\; AO\; ::\; AD\; :\; DO$, and by composition, $EO\; :\; AO\; ::\; AO\; :\; DO$, therefore $EO\; \backslash times\; OD\; =\; AO^2$. Hence, if the given points E and D (fig. 3), be so situated that $EO\; \backslash times\; OD\; =\; AO^2$, and at the same time $\backslash alpha\; :\; \backslash beta\; ::\; EA\; :\; AD\; ::\; EC\; :\; CD$, the problem admits of numberless solutions; and if either of the points D or E be given, the other point, and also the ratio which will render the problem indeterminate, may be found. Hence we have this porism:

"A circle ABC, and also a point D being given, another point E may be found, such that the two lines inflected from these points to any point in the circumference ABC,

shall have to each other a given ratio, which ratio is also to be found." Hence also we have an example of the derivation of porisms from one another, for the circle ABC, and the points D and E remaining as before, if, through D we draw any line whatever HDB, meeting the circle in B and H; and if the lines EB, EH, be also drawn, these lines will cut off equal circumferences BF, HG. Let FC be drawn, and it is plain from the foregoing analysis, that the angles DFC, CFB, are equal; therefore if OG, OB, be drawn, the angles BOC, COG, are also equal; and consequently the angles DOB, DOG. In the same manner, by joining AB, the angle DBE being bisected by BA, it is evident that the angle AOF is equal to AOH, and therefore the angle FOB to HOG; hence the arc FB is equal to the arc HG. It is evident that if the circle ABC, and either of the points DE were given, the other point might be found. Therefore we have this porism, which appears to have been the last but one of the third book of Euclid's Porisms. "A point being given, either within or without a circle given by position, if there be drawn, anyhow through that point, a line cutting the circle in two points; another point may be found, such, that if two lines be drawn from it to the points in which the line already drawn cuts the circle, these two lines will cut off from the circle equal circumferences."

The proposition from which we have deduced these two porisms, also affords an illustration of the remark, that the conditions of a problem are involved in one another in the porismatic or indefinite case; for here several independent conditions are laid down, by the help of which the problem is to be resolved. Two points D and E are given, from which two lines are to be inflected, and a circumference ABC, in which these lines are to meet, as also a ratio which these lines are to have to each other. Now these conditions are all independent of one another, so that any one may be changed without any change whatever in the rest. This is true in general; but yet in one case, viz. when the points are so related to another, that the rectangle under their distances from the centre is equal to the square of the radius of the circle, it follows, from the preceding analysis, that the ratio of the inflected lines is no longer a matter of choice, but a necessary consequence of this disposition of the points.

From what has been already said, we may trace the imperfect definition of a porism which Pappus ascribes to the later geometers, viz. that it differs from a local theorem, by wanting the hypothesis assumed in that theorem. Now, to understand this, it must be observed, that if we take one of the propositions called loci, and make the construction of the figure a part of the hypothesis, we get what was called by the ancient geometers, a local theorem. If, again, in the enunciation of the theorem, that part of the hypothesis which contains the construction be suppressed, the proposition thence arising will be a porism, for it will enunciate a truth, and will require to the full understanding and investigation of that truth, that something should be found, viz. the circumstances in the construction supposed to be omitted.

Thus, when we say, if from two given points, E, D, two straight lines EF, FD, are inflected to a third point F, so as to be to one another in a given ratio, the point F is in the circumference of a given circle, we have a locus. But

when conversely, it is said, if a circle ABC, of which the centre is O, be given by position, as also a point E; and if D be taken in the line EO, so that $EO\; \backslash times\; OD\; =\; AO^2$, and if from E and D the lines EF, DF be inflected to any point of the circumference ABC, the ratio of EF to DF will be given, viz. the same with that of EA to AD, we have a local theorem. Porism.

Lastly, when it is said, if a circle ABC be given by position, and also a point E, a point D may be found, such that if EF, FD be inflected from E and D to any point F in the circumference ABC, these lines shall have a given ratio to one another, the proposition becomes a porism, and is the same that has just now been investigated.

Hence it is evident, that the local theorem is changed into a porism, by leaving out what relates to the determination of D, and of the given ratio. But though all propositions formed in this way from the conversion of loci, are porisms, yet all porisms are not formed from the conversion of loci; the first, for instance, of the preceding, cannot, by conversion, be changed into a locus; therefore Fermat's idea of porisms, founded upon this circumstance, could not fail to be imperfect.

If the idea which we have given of these propositions be just, it follows, that they are to be discovered by considering those cases in which the construction of a problem fails, in consequence of the lines which by their intersection, or the points which by their position, were to determine the problem required, happening to coincide with one another. A porism may therefore be deduced from the problem to which it belongs, just as propositions concerning the maxima and minima of quantities are deduced from the problems of which they form limitations; and such is the most natural and obvious analysis of which this class of propositions admits.

Another general remark which may be made on the analysis of porisms is, that it often happens that the magnitudes required may all, or a part of them, be found by considering the extreme cases; but for the discovery of the relation between them, and the indefinite magnitudes, we must have recourse to the hypothesis of the porism in its most general or indefinite form; and must endeavour so to conduct the reasoning, that the indefinite magnitudes may at length totally disappear, and leave a proposition asserting the relation between determinate magnitudes only.

For this purpose accordingly Dr. Simson frequently employs two statements of the general hypothesis, which he compares together. This double statement, however, cannot be made without rendering the investigation long and complicated; nor is it even necessary, for it may be avoided by having recourse to simpler porisms, or to loci, or to propositions of the data. The porism which follows, is given as an example where this is done with some difficulty, but with considerable advantage, both with regard to the simplicity and shortness of the demonstration. It will be proper to premise the following lemma.

Let AB (fig. 4.) be a straight line, and D, L any two points in it, one of which D is between A and B; also let CL be any straight line. Then shall

For place CL perpendicular to AB, and through the points

Porism. A, C, B describe a circle, and let CL meet the circle again in E, and join AE, BE. Also draw DG parallel to CE, meeting AE and BE in H and G, and draw EK parallel to AB. Then, from the elements of geometry,

From the three equations now deduced, there results

two last propositions we have

But $DA\; \backslash times\; DH\; =\; \backslash text\{twice\; trian.\; ADH\}$, and $DB\; \backslash times\; DG\; =\; \backslash text\{twice\; trian.\; BDG\}$, and therefore $DA\; \backslash times\; DH\; +\; DB\; \backslash times\; DG\; =\; 2(\backslash text\{trian.\; ADH\}\; +\; \backslash text\{trian.\; BDG\})\; =\; 2(\backslash text\{trian.\; AEB\}\; +\; \backslash text\{trian.\; HEG\})\; =\; AB\; \backslash times\; LE\; +\; EK\; \backslash times\; HG$. Now it has been proved,